3.6.63 \(\int \frac {-b x^3+2 a x^8}{\sqrt [4]{-b x+a x^6} (-1-b x^5+a x^{10})} \, dx\)

Optimal. Leaf size=43 \[ \frac {2}{5} \tan ^{-1}\left (x \sqrt [4]{a x^6-b x}\right )-\frac {2}{5} \tanh ^{-1}\left (x \sqrt [4]{a x^6-b x}\right ) \]

________________________________________________________________________________________

Rubi [C]  time = 1.48, antiderivative size = 189, normalized size of antiderivative = 4.40, number of steps used = 12, number of rules used = 7, integrand size = 43, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.163, Rules used = {1593, 2056, 6728, 466, 465, 511, 510} \begin {gather*} -\frac {8 a x^4 \sqrt [4]{1-\frac {a x^5}{b}} F_1\left (\frac {3}{4};1,\frac {1}{4};\frac {7}{4};\frac {2 a x^5}{b-\sqrt {b^2+4 a}},\frac {a x^5}{b}\right )}{15 \left (b-\sqrt {4 a+b^2}\right ) \sqrt [4]{a x^6-b x}}-\frac {8 a x^4 \sqrt [4]{1-\frac {a x^5}{b}} F_1\left (\frac {3}{4};1,\frac {1}{4};\frac {7}{4};\frac {2 a x^5}{b+\sqrt {b^2+4 a}},\frac {a x^5}{b}\right )}{15 \left (\sqrt {4 a+b^2}+b\right ) \sqrt [4]{a x^6-b x}} \end {gather*}

Warning: Unable to verify antiderivative.

[In]

Int[(-(b*x^3) + 2*a*x^8)/((-(b*x) + a*x^6)^(1/4)*(-1 - b*x^5 + a*x^10)),x]

[Out]

(-8*a*x^4*(1 - (a*x^5)/b)^(1/4)*AppellF1[3/4, 1, 1/4, 7/4, (2*a*x^5)/(b - Sqrt[4*a + b^2]), (a*x^5)/b])/(15*(b
- Sqrt[4*a + b^2])*(-(b*x) + a*x^6)^(1/4)) - (8*a*x^4*(1 - (a*x^5)/b)^(1/4)*AppellF1[3/4, 1, 1/4, 7/4, (2*a*x
^5)/(b + Sqrt[4*a + b^2]), (a*x^5)/b])/(15*(b + Sqrt[4*a + b^2])*(-(b*x) + a*x^6)^(1/4))

Rule 465

Int[(x_)^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = GCD[m + 1,
n]}, Dist[1/k, Subst[Int[x^((m + 1)/k - 1)*(a + b*x^(n/k))^p*(c + d*x^(n/k))^q, x], x, x^k], x] /; k != 1] /;
FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && IntegerQ[m]

Rule 466

Int[((e_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> With[{k = Deno
minator[m]}, Dist[k/e, Subst[Int[x^(k*(m + 1) - 1)*(a + (b*x^(k*n))/e^n)^p*(c + (d*x^(k*n))/e^n)^q, x], x, (e*
x)^(1/k)], x]] /; FreeQ[{a, b, c, d, e, p, q}, x] && NeQ[b*c - a*d, 0] && IGtQ[n, 0] && FractionQ[m] && Intege
rQ[p]

Rule 510

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Simp[(a^p*c^q
*(e*x)^(m + 1)*AppellF1[(m + 1)/n, -p, -q, 1 + (m + 1)/n, -((b*x^n)/a), -((d*x^n)/c)])/(e*(m + 1)), x] /; Free
Q[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] && (IntegerQ[p] || GtQ[a
, 0]) && (IntegerQ[q] || GtQ[c, 0])

Rule 511

Int[((e_.)*(x_))^(m_.)*((a_) + (b_.)*(x_)^(n_))^(p_)*((c_) + (d_.)*(x_)^(n_))^(q_), x_Symbol] :> Dist[(a^IntPa
rt[p]*(a + b*x^n)^FracPart[p])/(1 + (b*x^n)/a)^FracPart[p], Int[(e*x)^m*(1 + (b*x^n)/a)^p*(c + d*x^n)^q, x], x
] /; FreeQ[{a, b, c, d, e, m, n, p, q}, x] && NeQ[b*c - a*d, 0] && NeQ[m, -1] && NeQ[m, n - 1] &&  !(IntegerQ[
p] || GtQ[a, 0])

Rule 1593

Int[(u_.)*((a_.)*(x_)^(p_.) + (b_.)*(x_)^(q_.))^(n_.), x_Symbol] :> Int[u*x^(n*p)*(a + b*x^(q - p))^n, x] /; F
reeQ[{a, b, p, q}, x] && IntegerQ[n] && PosQ[q - p]

Rule 2056

Int[(u_.)*(P_)^(p_.), x_Symbol] :> With[{m = MinimumMonomialExponent[P, x]}, Dist[P^FracPart[p]/(x^(m*FracPart
[p])*Distrib[1/x^m, P]^FracPart[p]), Int[u*x^(m*p)*Distrib[1/x^m, P]^p, x], x]] /; FreeQ[p, x] &&  !IntegerQ[p
] && SumQ[P] && EveryQ[BinomialQ[#1, x] & , P] &&  !PolyQ[P, x, 2]

Rule 6728

Int[(u_)/((a_.) + (b_.)*(x_)^(n_.) + (c_.)*(x_)^(n2_.)), x_Symbol] :> With[{v = RationalFunctionExpand[u/(a +
b*x^n + c*x^(2*n)), x]}, Int[v, x] /; SumQ[v]] /; FreeQ[{a, b, c}, x] && EqQ[n2, 2*n] && IGtQ[n, 0]

Rubi steps

\begin {align*} \int \frac {-b x^3+2 a x^8}{\sqrt [4]{-b x+a x^6} \left (-1-b x^5+a x^{10}\right )} \, dx &=\int \frac {x^3 \left (-b+2 a x^5\right )}{\sqrt [4]{-b x+a x^6} \left (-1-b x^5+a x^{10}\right )} \, dx\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \int \frac {x^{11/4} \left (-b+2 a x^5\right )}{\sqrt [4]{-b+a x^5} \left (-1-b x^5+a x^{10}\right )} \, dx}{\sqrt [4]{-b x+a x^6}}\\ &=\frac {\left (\sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \int \left (\frac {2 a x^{11/4}}{\sqrt [4]{-b+a x^5} \left (-b-\sqrt {4 a+b^2}+2 a x^5\right )}+\frac {2 a x^{11/4}}{\sqrt [4]{-b+a x^5} \left (-b+\sqrt {4 a+b^2}+2 a x^5\right )}\right ) \, dx}{\sqrt [4]{-b x+a x^6}}\\ &=\frac {\left (2 a \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \int \frac {x^{11/4}}{\sqrt [4]{-b+a x^5} \left (-b-\sqrt {4 a+b^2}+2 a x^5\right )} \, dx}{\sqrt [4]{-b x+a x^6}}+\frac {\left (2 a \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \int \frac {x^{11/4}}{\sqrt [4]{-b+a x^5} \left (-b+\sqrt {4 a+b^2}+2 a x^5\right )} \, dx}{\sqrt [4]{-b x+a x^6}}\\ &=\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{-b+a x^{20}} \left (-b-\sqrt {4 a+b^2}+2 a x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^6}}+\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^{14}}{\sqrt [4]{-b+a x^{20}} \left (-b+\sqrt {4 a+b^2}+2 a x^{20}\right )} \, dx,x,\sqrt [4]{x}\right )}{\sqrt [4]{-b x+a x^6}}\\ &=\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^4} \left (-b-\sqrt {4 a+b^2}+2 a x^4\right )} \, dx,x,x^{5/4}\right )}{5 \sqrt [4]{-b x+a x^6}}+\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{-b+a x^5}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\sqrt [4]{-b+a x^4} \left (-b+\sqrt {4 a+b^2}+2 a x^4\right )} \, dx,x,x^{5/4}\right )}{5 \sqrt [4]{-b x+a x^6}}\\ &=\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{1-\frac {a x^5}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-b-\sqrt {4 a+b^2}+2 a x^4\right ) \sqrt [4]{1-\frac {a x^4}{b}}} \, dx,x,x^{5/4}\right )}{5 \sqrt [4]{-b x+a x^6}}+\frac {\left (8 a \sqrt [4]{x} \sqrt [4]{1-\frac {a x^5}{b}}\right ) \operatorname {Subst}\left (\int \frac {x^2}{\left (-b+\sqrt {4 a+b^2}+2 a x^4\right ) \sqrt [4]{1-\frac {a x^4}{b}}} \, dx,x,x^{5/4}\right )}{5 \sqrt [4]{-b x+a x^6}}\\ &=-\frac {8 a x^4 \sqrt [4]{1-\frac {a x^5}{b}} F_1\left (\frac {3}{4};1,\frac {1}{4};\frac {7}{4};\frac {2 a x^5}{b-\sqrt {4 a+b^2}},\frac {a x^5}{b}\right )}{15 \left (b-\sqrt {4 a+b^2}\right ) \sqrt [4]{-b x+a x^6}}-\frac {8 a x^4 \sqrt [4]{1-\frac {a x^5}{b}} F_1\left (\frac {3}{4};1,\frac {1}{4};\frac {7}{4};\frac {2 a x^5}{b+\sqrt {4 a+b^2}},\frac {a x^5}{b}\right )}{15 \left (b+\sqrt {4 a+b^2}\right ) \sqrt [4]{-b x+a x^6}}\\ \end {align*}

________________________________________________________________________________________

Mathematica [F]  time = 0.27, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {-b x^3+2 a x^8}{\sqrt [4]{-b x+a x^6} \left (-1-b x^5+a x^{10}\right )} \, dx \end {gather*}

Verification is not applicable to the result.

[In]

Integrate[(-(b*x^3) + 2*a*x^8)/((-(b*x) + a*x^6)^(1/4)*(-1 - b*x^5 + a*x^10)),x]

[Out]

Integrate[(-(b*x^3) + 2*a*x^8)/((-(b*x) + a*x^6)^(1/4)*(-1 - b*x^5 + a*x^10)), x]

________________________________________________________________________________________

IntegrateAlgebraic [A]  time = 19.97, size = 43, normalized size = 1.00 \begin {gather*} \frac {2}{5} \tan ^{-1}\left (x \sqrt [4]{-b x+a x^6}\right )-\frac {2}{5} \tanh ^{-1}\left (x \sqrt [4]{-b x+a x^6}\right ) \end {gather*}

Antiderivative was successfully verified.

[In]

IntegrateAlgebraic[(-(b*x^3) + 2*a*x^8)/((-(b*x) + a*x^6)^(1/4)*(-1 - b*x^5 + a*x^10)),x]

[Out]

(2*ArcTan[x*(-(b*x) + a*x^6)^(1/4)])/5 - (2*ArcTanh[x*(-(b*x) + a*x^6)^(1/4)])/5

________________________________________________________________________________________

fricas [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x^8-b*x^3)/(a*x^6-b*x)^(1/4)/(a*x^10-b*x^5-1),x, algorithm="fricas")

[Out]

Timed out

________________________________________________________________________________________

giac [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, a x^{8} - b x^{3}}{{\left (a x^{10} - b x^{5} - 1\right )} {\left (a x^{6} - b x\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x^8-b*x^3)/(a*x^6-b*x)^(1/4)/(a*x^10-b*x^5-1),x, algorithm="giac")

[Out]

integrate((2*a*x^8 - b*x^3)/((a*x^10 - b*x^5 - 1)*(a*x^6 - b*x)^(1/4)), x)

________________________________________________________________________________________

maple [F]  time = 0.02, size = 0, normalized size = 0.00 \[\int \frac {2 a \,x^{8}-b \,x^{3}}{\left (a \,x^{6}-b x \right )^{\frac {1}{4}} \left (a \,x^{10}-b \,x^{5}-1\right )}\, dx\]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int((2*a*x^8-b*x^3)/(a*x^6-b*x)^(1/4)/(a*x^10-b*x^5-1),x)

[Out]

int((2*a*x^8-b*x^3)/(a*x^6-b*x)^(1/4)/(a*x^10-b*x^5-1),x)

________________________________________________________________________________________

maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \int \frac {2 \, a x^{8} - b x^{3}}{{\left (a x^{10} - b x^{5} - 1\right )} {\left (a x^{6} - b x\right )}^{\frac {1}{4}}}\,{d x} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x^8-b*x^3)/(a*x^6-b*x)^(1/4)/(a*x^10-b*x^5-1),x, algorithm="maxima")

[Out]

integrate((2*a*x^8 - b*x^3)/((a*x^10 - b*x^5 - 1)*(a*x^6 - b*x)^(1/4)), x)

________________________________________________________________________________________

mupad [F]  time = 0.00, size = -1, normalized size = -0.02 \begin {gather*} \int -\frac {2\,a\,x^8-b\,x^3}{{\left (a\,x^6-b\,x\right )}^{1/4}\,\left (-a\,x^{10}+b\,x^5+1\right )} \,d x \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(-(2*a*x^8 - b*x^3)/((a*x^6 - b*x)^(1/4)*(b*x^5 - a*x^10 + 1)),x)

[Out]

int(-(2*a*x^8 - b*x^3)/((a*x^6 - b*x)^(1/4)*(b*x^5 - a*x^10 + 1)), x)

________________________________________________________________________________________

sympy [F(-1)]  time = 0.00, size = 0, normalized size = 0.00 \begin {gather*} \text {Timed out} \end {gather*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate((2*a*x**8-b*x**3)/(a*x**6-b*x)**(1/4)/(a*x**10-b*x**5-1),x)

[Out]

Timed out

________________________________________________________________________________________